A motivic spectrum representing hermitian K-theory
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| Format: | Preprint |
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2024
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| _version_ | 1866913663033540608 |
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| author | Calmès, Baptiste Harpaz, Yonatan Nardin, Denis |
| author_facet | Calmès, Baptiste Harpaz, Yonatan Nardin, Denis |
| contents | We establish fundamental motivic results about hermitian K-theory without assuming that 2 is invertible on the base scheme. In particular, we prove that both quadratic and symmetric Grothendieck-Witt theory satisfy Nisnevich descent, and that symmetric Grothendieck-Witt theory further satisfies a projective bundle formula, as well as dévissage and A^1-invariance over a regular Noetherian base of finite Krull dimension. We use this to show that over a regular Noetherian base, symmetric Grothendieck-Witt theory is represented by a motivic E-infinity-ring spectrum, which we then show is an absolutely pure spectrum, answering a question of Déglise. As with algebraic K-theory, we show that over a general base, one can also construct a hermitian K-theory motivic spectrum, representing this time a suitable homotopy invariant and Karoubi-localising version of Grothendieck-Witt theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_15136 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A motivic spectrum representing hermitian K-theory Calmès, Baptiste Harpaz, Yonatan Nardin, Denis K-Theory and Homology Algebraic Geometry 19G38, 11E70, 14F42 We establish fundamental motivic results about hermitian K-theory without assuming that 2 is invertible on the base scheme. In particular, we prove that both quadratic and symmetric Grothendieck-Witt theory satisfy Nisnevich descent, and that symmetric Grothendieck-Witt theory further satisfies a projective bundle formula, as well as dévissage and A^1-invariance over a regular Noetherian base of finite Krull dimension. We use this to show that over a regular Noetherian base, symmetric Grothendieck-Witt theory is represented by a motivic E-infinity-ring spectrum, which we then show is an absolutely pure spectrum, answering a question of Déglise. As with algebraic K-theory, we show that over a general base, one can also construct a hermitian K-theory motivic spectrum, representing this time a suitable homotopy invariant and Karoubi-localising version of Grothendieck-Witt theory. |
| title | A motivic spectrum representing hermitian K-theory |
| topic | K-Theory and Homology Algebraic Geometry 19G38, 11E70, 14F42 |
| url | https://arxiv.org/abs/2402.15136 |